Lie algebra

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June 30, 2026

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A Lie algebra is a mathematical structure named after Norwegian mathematician Sophus Lie. It's a type of linear algebra where a binary operation, denoted by brackets [, ], satisfies specific properties. These properties include bilinearity, antisymmetry (meaning [A,A] = 0), and the Jacobi identity ([[A,B],C] + [[B,C],A] + [[C,A],B] = 0).1

Lie algebras are abstract settings for matrix algebra and are closely related to Lie groups, which describe motions that preserve certain properties of geometric figures. The theory of Lie algebras and Lie groups has found significant applications in geometry and physics, particularly in areas like particle physics and the study of fundamental forces.2

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mathematics
Mathematical physics and the theory of groups
…be called the classical complex Lie algebras. The simple Lie algebras, out of which all the others in the classification are made, were all representable as algebras of matrices, and, in a sense, Lie algebra is the abstract setting for matrix algebra. Connected to each Lie algebra there were a…
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…be called the classical complex Lie algebras. The simple Lie algebras, out of which all the others in the classification are made, were all representable as algebras of matrices, and, in a sense, Lie algebra is the abstract setting for matrix algebra. Connected to each Lie algebra there were a…
Read More